59.1.148 problem 150

Internal problem ID [9320]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 150
Date solved : Sunday, March 30, 2025 at 02:32:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)-x*(-x^2+1)*diff(y(x),x)+(x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x^{2}}{2}} x \left (\operatorname {Ei}_{1}\left (-\frac {x^{2}}{2}\right ) c_2 +c_1 \right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 35

ode=x^2*D[y[x],{x,2}]-x*(1-x^2)*D[y[x],x]+(1+x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-\frac {x^2}{2}} x \left (c_1 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )+2 c_2\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(1 - x**2)*Derivative(y(x), x) + (x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False